New limiting absorption and limit amplitude principles for periodic operators



We show a new limiting absorption and a new limit amplitude principle for periodic operators acting on functions defined on the whole \({\mathbb{R}^{d}}\) . For a differential operator \({\mathcal{L}}\) with periodic coefficients, we show the existence of the distributional limiting absorption solution of the Helmholtz equation (\({\mathcal{L}-\omega^{2}}\))v = g where \({\omega^{2}}\) is in the spectrum of \({\mathcal{L}}\) and regular in some sense. The limiting absorption solution is in some weighted L 2-space with respect to the spatial variable. Moreover, we regard \({\omega}\) as an additional variable, and we take the limit also in a L 2-norm with respect to \({\omega}\) . Based on this, we also derive a limit amplitude principle for the corresponding solution u of wave equation with time-harmonic right-hand side. In other words, we show in the same function spaces, the asymptotic behavior \({u(x) \sim e^{i\omega t}v^{-}(x)}\) as \({t \to +\infty}\) where v is the limiting absorption solution of the Helmholtz equation.

Mathematics Subject Classification (2010)

Primary 35B27 Secondary 78A40 35B40 35C99 


Limiting absorption principle Limit amplitude principle Periodic operator Floquet–Bloch transform Helmholtz equation Wave equation Asymptotic behavior 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Birman M.S., Yafaev D.R.: The scattering matrix for a perturbation of a periodic Schrödinger operator by decreasing potential. St. Petersburg Math. J. 6(3), 453–474 (1995)MathSciNetGoogle Scholar
  2. 2.
    Clarke F.H.: On the inverse function theorem. Pac. J. Math. 64(1), 97–102 (1976)CrossRefMATHGoogle Scholar
  3. 3.
    Dautray R., Lions J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5 Evolution Problems I. Springer, Berlin (2000)MATHGoogle Scholar
  4. 4.
    Eidus, D.M.: The Principle of Limiting Absorption, American Mathematical Society Translations, Series 2, Vol. 47, pp. 157–191 (1965)Google Scholar
  5. 5.
    Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)Google Scholar
  6. 6.
    Fliss, S.: Etude mathematique et numerique de la propagation des ondes dans un milieu periodique presentant un defaut. Ph.D. thesis, INRIA Paris-Rocquencourt, France (2009)Google Scholar
  7. 7.
    Gerard C., Nier F.: The Mourre theory for analytically fibered operators. J. Funct. Anal. 152, 202–219 (1998)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Grafakos L.: Classical Fourier Analysis. Springer, Berlin (2008)MATHGoogle Scholar
  9. 9.
    Hoang V.: The limiting absorption principle for a periodic semi-infinite waveguide. SIAM J. Appl. Math. 71(3), 791–810 (2011)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Joannopoulos J.D., Johnson S.G., Winn J.N., Meade R.D.: Photonic Crystals: Molding the Flow of Light, 2nd edition. Princeton University Press, Princeton, NJ (2008)Google Scholar
  11. 11.
    Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)MATHGoogle Scholar
  12. 12.
    Kuchment P.: Floquet Theory for Partial Differential Equations. Birkhäuser, Basel (1993)CrossRefMATHGoogle Scholar
  13. 13.
    Murata M., Tsuchida T.: Asymptotics of Green functions and the limiting absorption principle for elliptic operators with periodic coefficients. J. Math. Kyoto Univ. 46(4), 713–754 (2006)MATHMathSciNetGoogle Scholar
  14. 14.
    Tychonoff A.N., Samarski A.A.: Differentialgleichungen der mathematischen Physik. VEB Deutscher Verlag der Wissenschaften, Berlin (1959)MATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Institute for AnalysisKarlsruhe Institute of Technology (KIT)KarlsruheGermany

Personalised recommendations